On the moment limit of quantum observables, with an application to the balanced homodyne detection

TL;DR

This paper investigates the moment operators of quantum observables in balanced homodyne detection, demonstrating that high amplitude limits uniquely determine the measurement statistics and establishing conditions for constructing observables from sequences of measurements.

Contribution

It provides a rigorous analysis of the moment operators in balanced homodyne detection and shows how limits of these operators can define the measurement of quantum observables.

Findings

High amplitude limits determine the entire quadrature statistics.

Moment limits ensure weak convergence of probability measures.

Balanced homodyne detection's moment operators satisfy the necessary conditions.

Abstract

We consider the moment operators of the observable (i.e. a semispectral measure or POM) associated with the balanced homodyne detection statistics, with paying attention to the correct domains of these unbounded operators. We show that the high amplitude limit, when performed on the moment operators, actually determines uniquely the entire statistics of a rotated quadrature amplitude of the signal field, thereby verifying the usual assumption that the homodyne detection achieves a measurement of that observable. We also consider, in a general setting, the possibility of constructing a measurement of a single quantum observable from a sequence of observables by taking the limit on the level of moment operators of these observables. In this context, we show that under some natural conditions (each of which is satisfied by the homodyne detector example), the existence of the moment limits ensures that the underlying probability measures converge weakly to the probability measure of the limiting observable. The moment approach naturally requires that the observables be determined by their moment operator sequences (which does not automatically happen), and it turns out, in particular, that this is the case for the balanced homodyne detector.